Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player choosesone of 3 strategies: rock, paper, or scissoTS. Then, both players reveal their choices.The outcome is determined as follows. If both players choose the same strategy,neither player wins or loses anything. Otherwise:• "paper covers rock": if one player chooses paper and the other chooses rock,the player who chose paper wins and is paid 1 by the other player.• "scissors cut paper": if one player chooses scissors and the other chooses paper,the player who chose scissors wins and is paid 1 by the other player.• "rock breaks scissors": if one player chooses rock and the other player choosesscissors, the player who chose rock wins and is paid 1 by the other player.We can write the payoff matrix for this game as follows:rock paper scissors0.rock11-1раperscissors-1(a) Show that this game does not have a pure Nash equilibrium.(b) Show that the pair of mixed strategies xT= ( ) and yT= ) togetherare a Nash equilibrium.13

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Answered: Consider the 2-player, zero-sum game…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider a symmetric game with 10 players. Each player chooses among three strategies: x, y, and z. Let nx denote the number of players who choose x, ny denote the number of players who choose y, and nz denote the number of players who choose z. (So, nz = 10−nx−ny.) The payoff to a player from choosing strategy x is 10−nx (note that nx includes this player as well), strategy y is 13−2ny (again ny includes this player as well), and strategy z is 3. (a) Show that a Nash equilibrium must have at least one person choosing x and at least one person choosing y. (Hint: In a Nash equilibrium, no player can do better by doing something different.) b) Find all Nash equilibria.
In a Stackelberg game, what is the best response that follower firm 2 can make to the choice y1 already made by the leader, firm 1?
Consider a two-player game that is set up with two piles of stones. The two players are taking turns removing stones from one of the two piles. In each turn, a player must choose a pile and remove one stone or two stones from it. The player who removes the last stone (making both piles empty) wins the game. Show that if the two piles contain the same number n ∈ Z+ of stones initially, then the second player can always guarantee a win.
Consider a new card game between 2 players: Jim (player 1) and Angella (player 2) Jim is dealt two cards : ♡8 and ♠️9 . Angella is also dealt two cards: ♢6 and ♣️6 . Now, each of the players will play 1 card both at the same time. The payoff of Jim is 4 points if he plays a card of opposite color (red/black) than Angella, and otherwise his payoff is 10 points. The payoff of Angella is 3 points if the difference of the already played card numbers is greater than 4, otherwise her payoff is 2 points. 1. Find the Best Responses for Angella.2. Find all the Nash Equilibriums of the game (if any).
Given question is A basket of fruit is being assembled from apples, bananas, and oranges. What is the fewest number of fruit pieces that should be placed in the basket to ensure that there are at least 8 apples, 6 bananas, or 9 oranges?Your answer: The question is asking you to prepare for a worst-case scenario, not a best-case scenario. Imagine a game where you're not the one in charge of choosing the fruits - your opponent is in charge, and they don't want you to win. The opponent can get away with choosing 20 fruits (7 apples, 5 bananas, and 8 oranges) before the 21st fruit finally forces your condition to be satisfied. Therefore total required the fewest number of fruit pieces =(7+5+8)+1=21.My question:Why did you add 7+5+8 , instead of 8 apples, 6 bananas, or 9 oranges? The given is 8+6+9 not 7+5+8. Also why did you add +1? Here, you add 1: (7+5+8) +1 =21. Also, Is it pigeonhole formula? The pigeonhole is ⌈n/m⌉ not like this as far as i know. Thank you
In Section 5.5 we showed the following two-person, zero-sum game had a mixed strategy: Player B b1 b2 b3 a1 0 -1 2 Player A a2 5 4 -3 a3 2 3 -4 a. Which strategies are dominated? a1, a2, a3, b1, b2, b3 b. Does the game have a pure or mixed strategy?
In game theory, the strategy that has a higher payoff than any other strategy—no matter what theother player does—is also known as the:a) superior strategy.b) Nash equilibrium.c) dominant strategy.d) best possible outcome..
In a game, a participant is given three attempts to hit a ball. On each try, she either scores a hit, H, or a miss, M. The game requires that the player must alternate which hand she uses in successive attempts. That is, if she makes her first attempt with her right hand, she must use her left hand for the second attempt and her right hand for the third. Her chance of scoring a hit with her right hand is .7 and with her left hand is .4. Assume that the results of successive attempts are independent and that she wins the game if she scores at least two hits in a row. If she makes her first attempt with her right hand, what is the probability that she wins the game?
A game theorist is walking down the street in his neighborhood and finds $20. Just as he picks it up, two neighborhood kids, Jane and Tim, run up to him, asking if they can have it. Because game theorists are generous by nature, he says he’s willing to let them have the $20, but only according to the following procedure: Jane and Tim are each to submit a written request as to their share of the $20. Let t denote the amount that Tim requests for himself and j be the amount that Jane requests for herself. Tim and Jane must choose j and t from the interval [0,20]. If j + t ≤ 20, then the two receive what they requested, and the remainder, 20 - j - t, is split equally between them. If, however, j + t > 20, then they get nothing, and the game theorist keeps the $20. Tim and Jane are the players in this game. Assume that each of them has a payoff equal to the amount of money that he or she receives. Find all Nash equilibria.
Consider a "modified" form of "matching biased coins" game problem. The matching player is paid Rs.8 if the two coins turn both heads and Rs.12 if the coins turn both tails. The non-matching player is paid Rs.3 when the two coins do not match. Given the choice of being the matching or non-matching player, which one would you choose and what would be your strategy?.
What is the pure strategy Nash equilibrium outcome if there is one Is this a socially optimal outcome? If not, which outcome is preferred
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